A presentation of cnoidal wave theory for practical application

Wiegel, Robert L.

doi:10.1017/s0022112060001481

Cited by 112 publications

(43 citation statements)

References 4 publications

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“…(21) corresponds very well to cnoidal theory (Boussinesq, 1871;Korteweg and De Vries, 1895;Wiegel, 1960;Le Roux, 2007b).…”

Section: Data Output Areasupporting

confidence: 71%

“…(24) corresponds closely to cnoidal theory (Boussinesq, 1871;Wiegel, 1960;Demirbilek and Vincent, 2002).…”

Section: Data Output Areasupporting

confidence: 64%

“…(23). This was shown by Le Roux (2008a) to be in accordance with cnoidal theory, as graphically simplified by Wiegel (1960). The MCD w varies from 2 o L in deep water to 6 o L at breaking depth, and is used to calculate the horizontal water particle semi-excursion (A hwz ) and velocity under the wave crest (U chwz ) in any water depth and at any distance z from the SWL (Eqs.…”

Section: Theoretical and Empirical Basis Of Equations Used In The Promentioning

confidence: 70%

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WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

et al. 2010

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The generation and growth of waves in deep water is controlled by winds blowing over the sea surface. In fully developed sea states, where winds and waves are in equilibrium, wave parameters may be calculated directly from the wind velocity. We provide an Excel spreadsheet to compute the wave period, length, height and celerity, as well as horizontal and vertical particle velocities for any water depth, bottom slope and distance below the reference water level. The wave profile and propagation can also be visualized for any water depth, modeling the sea surface change from sinusoidal to trochoidal and finally cnoidal profiles into shallow water. Bedload entrainment is estimated under the wave crest and trough, respectively, using the horizontal water particle velocity at the top of the boundary layer.The calculations are programmed in an Excel file called WAVECALC, which is available online to authorized users. Although many of the recently published formulas are based on theoretical arguments, the values agree well with several existing theories and limited field and laboratory observations. WAVECALC is a user-friendly program intended for sedimentologists, coastal 2 engineers and oceanographers, as well as marine ecologists and biologists. It provides a rapid means to calculate many wave characteristics required in coastal and shallow marine studies and can also serve as an educational tool.

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“…(21) corresponds very well to cnoidal theory (Boussinesq, 1871;Korteweg and De Vries, 1895;Wiegel, 1960;Le Roux, 2007b).…”

Section: Data Output Areasupporting

confidence: 71%

“…(24) corresponds closely to cnoidal theory (Boussinesq, 1871;Wiegel, 1960;Demirbilek and Vincent, 2002).…”

Section: Data Output Areasupporting

confidence: 64%

Section: Theoretical and Empirical Basis Of Equations Used In The Promentioning

confidence: 70%

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WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

et al. 2010

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“…Stokes solution (Fenton, 1985), a 2 nd order Cnoidal model (Wiegel, 1960), a higher order solitary wave theory (Munk, 1949) as well as stream function theory (Dean, 1965).…”

Section: Comparison Of Fluid Kinematics With Theoretical Formulationsmentioning

confidence: 99%

Composite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematics

Heller

Bruggemann

Spinneken

et al. 2016

Coastal Engineering

132

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk 05.01.2016, accepted for publication in Coastal Engineering 109(3), 20-41 (doi:10.1016/j.coastaleng.2015

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“…Putting the modulus µ at 0.99 and H/d to 0.01 in (2.17), the wave period T p is then equal to 191.6 s and the wavelength 4246 m according to cnoidal wave theory (Wiegel 1960). Figure 4 shows the run-up height as a function of time for various bed slopes.…”

Section: Series Solutionmentioning

confidence: 99%

Currents induced by long waves propagating towards a beach over a wavy bed

Kyotoh

Fujii²,

2000

J. Fluid Mech.

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For the understanding of longshore currents along a natural beach, the effects of bottom unevenness are considered to be important, especially for the flow in the swash zone. Currents in the swash zone are strongly influenced by the bed slope because the effect of gravity overwhelms the effect of the depth change. In the present paper, we investigate these effects and focus on waves propagating from offshore over a flat ocean basin of constant depth to a beach with a sloping wavy bottom. The waves are incident at a small angle to the beach normal, and the bed slope in the alongshore direction is varied slowly. To simplify the problem, only cnoidal waves and solitary waves are considered and the bed level is varied sinusoidally in the longshore direction.A perturbation method is applied to the two-dimensional nonlinear shallow water equation (two-dimensional NLSWE) for the wave motion in order to generate a more simplified model of wave dynamics consisting of a one-dimensional NLSWE for the direction normal to the beach and an equation for the alongshore direction. The first equation, the one-dimensional NLSWE, is solved by Carrier & Greenspan's transformation. The solution of the second one is found by extending Brocchini & Peregrine's solution for a flat beach. Two methods for the solution of the onedimensional NLSWE are introduced in order to get a solution applicable to largeamplitude swash motions, where the amplitude is comparable to the beach length. One is the Maclaurin expansion of the solution around the moving shoreline, and the other is Riemann's representation of the solution, which exactly satisfies the onedimensional NLSWE and the boundary conditions. After doing a consistency check by confirming that Riemann's method, a numerical solution, agrees with the exact solution for an infinitely long, sloping beach, we assumed that the Maclaurin series solution can also describe wave motion in the swash zone properly not only for this model but also for our 'wavy', finite beach model.The solution obtained from the Maclaurin series is then plugged into the equation for the alongshore direction to calculate the shore currents induced by wave run-up and back-wash motions, where a 'weakly two-dimensional solution' is derived from geometrical considerations. The results show that since the water depth near the shoreline is comparable to the bed level fluctuations, the flow is strongly affected by the bed unevenness, leading to recognizable changes in shoreline movement and the time-averaged velocity and the mass flux of the flow in the swash zone. More specifically, the inhomogeneity of the alongshore mass flux generates offshore currents because of the continuity condition for the fluid mass.

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A presentation of cnoidal wave theory for practical application

Cited by 112 publications

References 4 publications

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

Composite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematics

Currents induced by long waves propagating towards a beach over a wavy bed

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